Advances in Difference Equations
Volume 2009 (2009), Article ID 169321, 14 pages
doi:10.1155/2009/169321

The Existence of Positive Solutions for Third-Order p-Laplacian m-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales

Abstract

We study the following third-order p-Laplacian m-point boundary value problems on time scales (ϕp(uΔ∇))∇+a(t)f(t,u(t))=0, t∈[0,T]Tκ, u(0)=∑i=1m−2biu(ξi), uΔ(T)=0, ϕp(uΔ∇(0))=∑i=1m−2ciϕp(uΔ∇(ξi)), where ϕp(s) is p-Laplacian operator, that is, ϕp(s)=|s|p−2s, p>1, ϕp−1=ϕq,1/p+1/q=1, 0<ξ1<⋯<ξm−2<ρ(T). We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term f(t,u) is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.